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Graphing Linear Equations/Transcript
Transcript Title text reads, The Mysteries of Life with Tim and Moby. On-screen, an orange robot, Moby, stands in a field next to a beehive. Moby’s friend, a boy named Tim, stands nearby. Moby looks annoyed and beeps. TIM: Well, what did you expect from only ten bees? You’re gonna need to get some more bees if this honey business is going to go anywhere. On-screen, a letter appears. Text reads as Tim narrates: Dear Tim and Moby, How do I graph a linear equation? From, Mark Moby bangs on the beehive with his fist. TIM: Graphing is a good way to visualize what happens when you plug different numbers into an equation. A label appears, reading equation. Moby beeps. TIM: Well, an equation is a comparison of two sets of numbers with an equals sign, like 3 times 4 equals 12. An equation appears, reading, 3 times 4 equals 12. Above it, a label reads, linear equation. TIM: A linear equation is sort of like that, only we don’t know all of the numbers. We use placeholders called variables to represent these unknown amounts. An equation appears, reading, y equals 2x plus 7. TIM: Any letter you see inside an equation is a variable. A label above it reads, variables. X and y are used a lot, but you can really use any letter you want. In the equation, the x and the y turn red. Then they change to other variables, so the equation reads, r equals 2d plus 7. Next, it reads, f equals 2g plus 7. Finally, it reads, p equals 2v plus 7. Moby beeps questioningly. TIM: Oh, it has everything to do with bees, my friend. A linear equation will show us how the number of bees in your hive is related to how much honey you’ll get. Simply put, a linear equation is an equation whose graphical form is a straight line. On-screen, a coordinate plane is displayed. Two diagonal lines appear on the plane. One goes from the top left corner of the graph to the bottom right corner. It's marked, y equals negative x plus 5. The second goes from the bottom left corner of the graph to the top right corner. It's marked, y equals 0.5x plus 2. A label appears, reading, linear equation. TIM: Many linear equations are also functions. They describe the relationship between an output value and an input value. In our case, the output is honey, and the input is bees. A label appears, reading, functions. On-screen, drops of honey appear on the left, and a swarm of bees appears on the right. TIM: How many bees do you have now? Moby frowns and beeps. TIM: Okay, 10 bees. On-screen, 10 bees appear on the right side of the screen. TIM: And how much honey did you get after a week? Moby holds up a tiny glass of honey. TIM: 2 measly milliliters of honey. On-screen, two drops of honey appear to the left of the 10 bees. A ratio, reading, 2 over 10, appears between the honey and the bees. TIM: So 10 bees are producing 2 milliliters of honey per week. That number is a ratio, and we can reduce it like we'd reduce a fraction. For every 5 bees, 1 milliliter of honey is produced. On-screen, the 10 bees change to 5 bees, and the 2 drops of honey change to 1 drop. The ratio 1 over 5. Moby beeps and glares at his tiny glass of honey. TIM: We can express that with the equation, y equals x times one-fifth. Y is the amount of honey, x is the number of bees, and one-fifth is the rate at which the bees make the honey: 1 milliliter per 5 bees. An equation appears, reading: y equals x times one-fifth. Moby beeps again. TIM: See? Y is the output of our linear function, and x is the input! By plugging in different numbers of bees, x, we can find out how much honey we’ll get. On-screen, new equations appear beneath y equals x times one-fifth. The new equations read, 2 equals 10 times one-fifth, 4 equals 20 times one-fifth, and 8 equals 40 times one-fifth. Moby beeps. TIM: Well, we can use our trusty coordinate plane, a two-dimensional representation of space. A label reads, coordinate plane. On-screen, a coordinate plane appears. It's a grid divided into four sections by a horizontal line and a vertical line. The spot where the axees meet is labeled, 0 comma 0, in parentheses. This spot is called the origin. TIM: That space is split up by an x-axis and a y-axis. On-screen, the horizontal axis is labeled with an x and the vertical axis is labeled with a y. TIM: And those axees correspond to the x and y from our equation! On-screen, a bee appears next to the x-axis; a honey drop appears near the y-axis. TIM: We can use the x and y values from our equation to map points on the plane. Check it out: we have 10 bees. On-screen, a hash-mark appears on the x-axis, to the right of the origin. It's labeled with the number 10. TIM: And we know that that gives us 2 milliliters of honey. On-screen, a hash-mark appears on the y-axis, above the origin. It's labeled with the number 2. The coordinates of that point are 10 and 2. We show that as an ordered pair, in the format x comma y in parentheses. On-screen, a dotted line starts from the 2 on the y-axis, extending to the right. Another dotted line starts from the 10 on the x-axis, extending upward. The lines meet at a point 10 digits to the right of the origin, and 2 digits above the origin. That point is labeled, 10 comma 2 in parentheses. A label appears on the graph, reading, ordered pair. Moby beeps. TIM: Well, let’s plug a different number into our equation. Say we have 50 bees! That gives us the coordinates of 50 and 10. An equation reads, y equals x times one-fifth. The x is replaced by the number 50, so that it now reads, y equals 50 times one-fifth. Then the y is replaced by the number 10, so the equation reads, 10 equals 50 times one-fifth. Beneath the equation, an ordered pair appears, reading, 50 comma 10 in parentheses. TIM: Now, we’ve got two points graphed out. On-screen, 50 is marked on the x-axis, an 10 is marked on the y-axis. A dotted line extends from each mark. They meet at a point that's 50 digits to the right of the origin, and 10 digits above. It's labeled with the ordered pair, 50 comma 10 in parentheses. TIM: And since linear functions graph as straight lines, two points is really all we need. On-screen, a ruler appears between the points at 10 comma 2 and 50 comma 10. A pencil draws a diagonal line through the two points. The line extends to the edge of the coordinate plane in both directions. TIM: Any point along that line will give you the input and output values of our linear function. On-screen, various points appear along the diagonal line, including 20 comma 4 and 30 comma 6. TIM: See? 150 bees will get us… 30 milliliters of honey! On-screen, 150 is marked on the x-axis, an 30 is marked on the y-axis. A dotted line extends from each mark. They meet at a point that's 150 digits to the right of the origin, and 30 digits above. It's labeled with the ordered pair, 150 comma 30 in parentheses. Moby beeps. TIM: Well, because it’s visual, a graph can communicate information a lot faster than an equation. On-screen, Tim’s finger runs along the diagonal line that connects all the ordered pairs he's mentioned. TIM: Without looking at the coordinates, we can see that this line isn’t very steep. So with just a glance, we know that it’s going to take a lot of bees to make a good amount of honey. On-screen, a fuel truck backs up next to Moby and the beehive. The truck driver gets out and begins pumping bees into the hive with a big hose. TIM: How many bees did you order? Moby beeps, looking pleased with himself. Tim’s jaw drops. TIM: 700,000 bees!? Can they even fit in that one hive? On-screen, the truck drives off. Moby opens the top of the hive and looks inside. He comes up with his face covered in bees. Moby lumbers toward Tim, waving his arms and beeping. TIM: Auuuugh! Category:BrainPOP Transcripts Category:BrainPOP Math Transcripts